Central binomial coefficients from (1 − 4x)^(−1/2)

Derivatives. $f'(x)=2(1-4x)^{-3/2}$, $f''(x)=2\cdot6(1-4x)^{-5/2}$, $f'''(x)=2\cdot6\cdot10(1-4x)^{-7/2}$. By induction,
$$f^{(n)}(x)=\prod_{k=0}^{n-1}(4k+2)\;(1-4x)^{-(2n+1)/2},$$
since differentiating $(1-4x)^{-(2n+1)/2}$ multiplies by $\frac{2n+1}2\cdot4=4n+2$.

Coefficient. Therefore
$$\frac{f^{(n)}(0)}{n!}=\frac{2\cdot6\cdot10\cdots(4n-2)}{n!}=\frac{2^n\cdot1\cdot3\cdot5\cdots(2n-1)}{n!}.$$
Using $1\cdot3\cdots(2n-1)=\dfrac{(2n)!}{2^nn!}$, this equals $\dfrac{(2n)!}{n!\,n!}=\dbinom{2n}n$.

First terms. $(1-4x)^{-1/2}=1+2x+6x^2+20x^3+70x^4+\cdots$ ✓.

Remark. This is the generating function of the central binomial coefficients. Applying the same idea to $\sqrt{1-4x}$ produces the Catalan numbers.